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Among the ways to define Chern classes for vector bundles in algebraic topology, defining them by pulling back cohomology classes from the classifying space $BU(n)$ is one elegant way to do so.

In algebraic geometry there is no classifying scheme, but one can define the stack $M$ of vector bundles over schemes. I don't know anything about this stack.

Question: Is there a reasonable notion of Chow ring for $M$ so that the algebraic chern classes are obtained by pulling back cycles from $M$ along classifying maps? Or is this for some reason the wrong question?

(By algebraic Chern classes I mean what we obtain by formally inverting the generating function built from the Segre classes of a vector bundle, which are an endomorphism of the chow ring obtained by pulling back, intersecting with the canonical divisor in the bundle some number of times, and then pushing forward. I think these algebraic Chern classes can also be defined by twisting a vector bundle until it has enough sections, and then looking at the vanishes scheme of some the minors.)

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  • $\begingroup$ Interestingly, although classifying spaces of algebraic groups are not defined, their Chow rings are: math.ucla.edu/~totaro/papers/public_html/chowBG.pdf I don't think this answers your question, though. $\endgroup$ – Nefertiti Oct 10 '16 at 9:45
  • $\begingroup$ For me this does not look like a reasonable analog. Let call the space $X$, then 1. $M$ depends on $X$, but $BGL(n)$ does not. 2. What is map from $X$ to $M$? 3. In all examples I know, we need first fix Chern classes as discrete parameters of vector bundles and then study moduli stack as continuous parameters that are left. If we don't do this we get a disjoint union of moduli stacks for different values of Chern classes and you can determine Chern classes by saying in which component is the point representing the vector bundle. But this is recursive. $\endgroup$ – Alex Oct 10 '16 at 15:41
  • $\begingroup$ You can construct a spectra in the category of (P^1-stable) motivic spaces that coincides with usual (vector bundle) K-theory for smooth schemes, so there's a classifying space (constructed exactly in the same way by picking Grassmanians). Furthermore they're compatible with "motivic" Chern classes. I'm not sure if these motivic Chern classes are compatible with the non-étale version of Chern classes though. $\endgroup$ – user40276 Oct 13 '16 at 19:59
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    $\begingroup$ @Nefertiti: Yes it does answer the question. And yes, classifying spaces are defined, namely as quotient stacks $[Spec(k)/G]$, where you take the trivial action of $G$ on $Spec(k)$. These are Artin stacks of dimension $-dim(G)$. Their Chow groups are the $G$-equivariant Chow groups of $Spec(k)$. And you can again define Chern classes by pulling back generators of the Chow ring of $[Spec(k)/GL(n)]$. $\endgroup$ – Rieux Oct 13 '16 at 20:01
  • $\begingroup$ @Rieux: fair enough. I guess a more accurate version of my comment is that Totaro's paper defines the Chow ring of a classifying space in a way that makes sense without defining the classifying space itself. $\endgroup$ – Nefertiti Oct 14 '16 at 10:31

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